How to find an eigen space of this 2 by 2 matrix
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How to find an eigen space of this 2 by 2 matrix

[From: ] [author: ] [Date: 12-06-19] [Hit: ]
so that every vector v becomes -v so that every vector is an eigenvector with eigenvalue -1.Thus the eigenspace corresponding to the eigenvalue of -1 is R2.......
[-1 0]
[0 -1]

I want to find the eigenspace corresponding to the eigenvalue -1

-
If A is the given matrix then the characteristic polynomial is given by |A - tI| where t is a parameter and I is the identity matrix of M2,2.

Characteristic polynomial = (-1 - t)(-1 - t) = (1 + t)^2.
So we have a single eigenvalue of -1 as stated.
Use row reduction:

[0 0]
[0 0]

This means that there are no restrictions on x, y - that is to say ALL vectors of the form (x, y) satisfy the requirements of the eigenvalue. This is because the above matrix makes every vector negative, so that every vector v becomes -v so that every vector is an eigenvector with eigenvalue -1.

Thus the eigenspace corresponding to the eigenvalue of -1 is R2.
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